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# The sum of all the digits of the integers from 18 to 21 incl

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Re: The sum of all the digits of the integers from 18 to 21 incl [#permalink]
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What we want is 0+(1+2+3+....+9)+(1+1+1+2+1+3...1+9)+.....(9+1+9+2..+9+9)

1+2+..+9=(9*10)/2=45
Similarly 11+12+...+19 will have 10 1s and again a sequence of (1+2+3+..+9). 20 series will be (2+0+2+1..2+9)=20+(1+2+3..+9)

Extrapolating this, we will get 45+(10+45)+(20+45)+....(90+45) = 45+(9*45)+(10+20+30+...+90)=450+10(1+2+..+9)=
450+10(45)=900.

Not sure if there is a better way to do this.
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Re: sum of 0-99 [#permalink]
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Simple counting is what needed in this question
First of all we need to count how many 1 's are present from 0-99
i.e 1, 11-19, 21, 31, 41, 51, 61, 71, 81 ,91.... thus total of 20 1's are present
Thus total will be 1 * 20 = 20

Similarly there will be 20 2's and 20 3's ....... 20 9's

Therefore of each number will be 2*20 = 40, 3*20 = 60 ....... 9*20 = 180

Thus total of 20+40+60+80+100+120+140+160+180 = 900

Answer is "C"
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Re: sum of 0-99 [#permalink]
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This is my method

Sum of the digits from 1 to 9 = Sum of consecutive numbers = first plus last divided by two X number of terms = 5 X 9 = 45

Number of times the digits from one to nine appear in the sequence from 1 to 99 = 10

First part of the answer = 45 X 10 = 450

Number of times the number one (two..nine) appears in the sequence from 1 to 99 in the units digit = 10
1 X 10 = 10
2 X 10 = 20
.....
Sum of consecutive numbers = First Plus Last / 2 X Number of terms

90 + 10 = 100 / 2 = 50 X 9 = 450 Second part of the answer

Final answer 450 + 450 = 900

It worked for me...

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Re: sum of 0-99 [#permalink]
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1+2+3+4+5+6+7+8+9
(1+0)+(1+1)+(1+2)+(1+3)+(1+4)+(1+5)+(1+6)+(1+7)+(1+8)+(1+9)
(2+0)+(2+1)+(2+2)+(2+3)+(2+4)+(2+5)+(2+6)+(2+7)+(2+8)+(2+9)
and so on..

You don't have to write the above in the test... just quickly visualize it. What do we see here?

1+2+... +9, which adds to 45 appears 10 times
=> 45*10 = 450 -------------- (1)

also, we have 10 1's, 10 2's and so on
=> 10 (1 + 2 +...+ 9)
=> 10 * 45 = 450 ------------- (2)

(1) + (2) = 900

Hope this helps!
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Re: sum of 0-99 [#permalink]
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think of all numbers between 0-99 as double-digit numbers(1=01, 2=02 and so on)

sum of unit digits
digits 1-9 repeat 10 times (01 to 09, 11 to 19 .... 91 to 99)

sum of unit digits $$= 10*[\frac{9}{2}*(1+9)] = 450$$

sum of tens digits
digits 1-9 repeat 10 times (10 to 19, 20 to 29 .... 90 to 99)

sum of unit digits $$= 10*[\frac{9}{2}*(1+9)] = 450$$

total = 450 + 450 = 900

Ans: C
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Re: sum of 0-99 [#permalink]
1
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0+1+2+...+9 = x = 45

now we have,
unit's digit sum from 0-99 = 10 * x
ten's digit sum from 0-99 = 10* x

hence total is 20*x = 20 * (45) = 900. Hence C.
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Re: sum of 0-99 [#permalink]
we can see the pattern here

units digit : 1,2,3,4,5,6,7,8,9 repeated 10 times

=> unit digit sum = 10*(1+2+3+4+5+6+7+8+9)

tens digit : 1 repeated 10 times, 2 repeated 10 times...9 repeated 10 times

=> tens digit sum = 10*(1+2+3+4+5+6+7+8+9)

so their sum = 20*(1+2+3+4+5+6+7+8+9) = 900

Answer is C.
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Re: The sum of all the digits of the integers from 18 to 21 incl [#permalink]
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shrive555
The sum of all the digits of the integers from 18 to 21 inclusive is 24 (1+8+1+9+2+0+ 2+1=24). What is the sum of all the digits of the integers from 0 to 99 inclusive?

A. 450
B. 810
C. 900
D. 1000
E. 1100

1+2+3+4...+9 sum of all digits is 45

Each digit will appear ten times in the units digit = 10*45= 450
Each digit will appear once in the tens digit = 1*45*10 = 450 (The ten is cause its in the tens digit)
Total= 900

Answer is C

Hope it helps

Gimme Kuddos

Cheers
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Re: The sum of all the digits of the integers from 18 to 21 incl [#permalink]
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Expert Reply
Sum from 0 - 9 = 45
Sum from 10 - 19 = (1x10 + 45) = 55
Sum from 20 - 29 = (2x10 + 45) = 65
.
.
.
Sum from 90 - 99 = (9x10 + 45) = 135

Total sum = 55 + 65 ..... 135 = 10(45+135)/2 = 900
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Re: The sum of all the digits of the integers from 18 to 21 incl [#permalink]
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0 to 9 in units place appear 10 times, so 10x0 + 10x1 + ............ + 10x9 = 450
0 to 9 in tens place appear 10 times, so 10x0 + 10x1 + ............ + 10x9 = 450

450+450=900 = Answer = C
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The sum of all the digits of the integers from 18 to 21 incl [#permalink]
shrive555
The sum of all the digits of the integers from 18 to 21 inclusive is 24 (1 + 8 + 1 + 9 + 2 + 0 + 2 + 1 = 24). What is the sum of all the digits of the integers from 0 to 99 inclusive?

A. 450
B. 810
C. 900
D. 1000
E. 1100

Many of the over-mentioned require a good number sense, which I do not posses, so I'll show you my "layman" approach

We want the sum of the digits from 0 to 99, so I approximated:
0-9 -> 45 -> (9+0)*10/2
40-49 -> 85 (13+4)*10/2
90-99 -> 135 (18+9)*10/2

We can see at a glance that the "weight" goes up as the numbers go up (meaning the difference between 85 and 45 is 40, while 135-85 is 50, this means that the second part of this sequence carries more weight for our result), so we know that the final answer has to be more than 850 (85*10) but close to it, and that's just 900: the answer is C.

Kudos if you like.

Den.
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Re: The sum of all the digits of the integers from 18 to 21 incl [#permalink]
VeritasKarishma
In the first 99 numbers, every digit appears 20 times. (Think 2 - It appears 10 times in units digit and 10 times in tens digit)

So all you need is (1 + 2 + 3 + ... + 9) x 20 = 45 x 20 = 900

Note: Remember 1 + 2 + 3 +...n = n(n+1)/2

VeritasKarishma, your solution to this is great, but do you not think that there must be reason to give 18 to 21 = 24. Is there a possibility to use this and reach to the answer earlier?
Usually Questions mentions "such as" or "for example" if they add info just to make us understand the question better!!!

Since this question is from Magoosh, mikemcgarry, can you provide the official solution?
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Re: The sum of all the digits of the integers from 18 to 21 incl [#permalink]
A different approach that I havent seen on this question

notice if we sum up 0-9 we get 1+2+...9 = 45

for digits 10-19 we get 1+2+3 +.....19 = 55

for digits 20-29 we get 2+3+.....29 = 65.

In fact as we increase tens place digit by one for each row of our sum, the sum will increase by 10

so the sum = (45+55+65+75+85+95+105+115+125+135) = 900
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Re: The sum of all the digits of the integers from 18 to 21 incl [#permalink]
﻿There are a total of 100 numbers in the range of 0 to 99.
﻿This includes a total of 10 one-digit numbers and 90 two-digit numbers.
﻿A total of 10*1 + 90*2 = 190 digits needs to be added.
﻿But of the 190 digits, 0 appears a total of 10 times.
﻿(0, 10, 20, 30, 40, 50, 60, 70, 80, 90)
﻿The other digits repeat themselves a total of 20 times each.
﻿The sum is given by :
﻿20*(1+2+3+4+5+6+7+8+9)
﻿20*(45) = 900
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Re: The sum of all the digits of the integers from 18 to 21 incl [#permalink]
Hello,

I started from the first and the last digits:

0 + 9+9 = 18
1 + 9+8 = 18
2 + 9+7 = 18
3 + 9+6 = 18
.
.
.
4+8 + 5+1 = 18
4+9 + 5+0 = 18

so we have 50 lines with 18 as a sum, so I just multiply 18*50= 9*100 (double and half) = 900
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Re: The sum of all the digits of the integers from 18 to 21 incl [#permalink]
KarishmaB
shrive555
The sum of all the digits of the integers from 18 to 21 inclusive is 24 (1+8 + 1+9 + 2+0 + 2+1 = 24). What is the sum of all the digits of the integers from 0 to 99 inclusive?

450
810
900
1000
1100
0 to 99 could be considered 100 two digit numbers(00, 01, 02....99) since when 0 is added, it doesn't change the sum. To write these 100 numbers, 200 digits will be used. Since there are 10 digits (0 to 9), each digit will appear 20 times.

So total sum = 20*0 + 20*1 + 20*2 + 20*3 +.....20*9 = 20*9*10/2 = 900
­Pls explain how for 100 numbers - 200 digits will be used part
Re: The sum of all the digits of the integers from 18 to 21 incl [#permalink]
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